Mixed displacement and couple stress finite element method for anisotropic centrosymmetric materials

Highlights

• Novel $C^0$ mixed finite element method is developed for consistent couple stress theory.

• Weak form is written in terms of displacement and couple-stress polar vectors.

• General method is applied to anisotropic centrosymmetric materials for first time.

• Examples consider cubic, hexagonal, trigonal, tetragonal single crystal materials.

• Stress concentration factor and non-dimensional stiffness studied in model problems.

Abstract

The classical theory of elasticity is an idealized model of a continuum, which works well for many engineering applications. However, with careful experiments one finds that it may fail in describing behavior in fatigue, at small scales and in structures having high stress concentration factors. Many size-dependent theories have been developed to capture these effects, one of which is the consistent couple stress theory. In this theory, couple stress ${\mu }_{ij}$ is present in addition to force stress ${\sigma }_{ij}$ and its tensor form is shown to have skew symmetry. The mean curvature ${\kappa }_{ij}$, which is defined as the skew-symmetric part of the gradient of rotations, is the correct energy conjugate of the couple stress. This mean curvature ${\kappa }_{ij}$ and strain $e_{ij}$ together contribute to the elastic energy. The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method. A fully displacement based finite element method for couple stress elasticity requires $C^1$ continuity. To avoid this, a mixed formulation is presented with primary variables of displacements $u_i$ and couple stress ${\mu }_i$ vectors, both of which require only $C^0$ continuity. Centrosymmetric classes of materials are considered here for which force stress and strain are decoupled from couple stress and mean curvature in the constitutive relations. Details regarding the numerical implementation are discussed and the effect of couple stress elasticity on anisotropic materials is examined through several computational examples.

Introduction

Classical continuum mechanics predicts the behavior of structures under loads reasonably well at macro scale, but careful experiments have shown that it deviates in capturing behavior of materials at micro scale. Molecular mechanics theory can be used to capture these small scale behaviors but is too computationally intensive to use for practical applications. Hence many size dependent continuum mechanics theories were developed in the past to bridge the gap between problems in the classical and molecular regimes. In classical elasticity theory, forces are transmitted at an infinitesimal element surface as tractions or more specifically force tractions. On the other hand, in size dependent theories, moments are transmitted on an infinitesimal element surface as moment or couple tractions in addition to force tractions. These force and moment tractions can then be represented by tensorial (force) stresses and couple stresses on infinitesimal elements. Correspondingly new measures of deformation, such as curvatures, are introduced in addition to strains.

Couple stresses were initially proposed by Voigt (1887), but the first mathematical model was presented by Cosserat and Cosserat (1909). Displacements and independent rotations, known as microrotations, were used as the kinematical quantities. Their work was further revived by Mindlin (1964), Eringen (1999), Nowacki (1986) and Chen and Wang (2001). These theories are popularly known today as the micropolar theories.

Another branch of theories, known as second gradient or strain gradient theories were developed by Mindlin and Eshel (1968), Yang et al. (2002) and Lazar et al. (2005). These involve gradients of strains, rotations and their various combinations all originating from the displacement field to avoid the microrotations.

One other branch of theories based on Voigt (1887) was developed by Toupin (1962), Mindlin and Tiersten (1962) and Koiter (1964) in which displacements and macrorotations were taken as the kinematical quantities. These macrorotations are the continuum mechanical rotations, which are defined as one half the curl of displacements. Finally the curvatures are defined as gradient of these macrorotations. But these theories had some indeterminacy in the couple stress and force stress tensors due to the limited number of relations. Recently, Hadjesfandiari and Dargush (2011) resolved this indeterminacy and showed the couple stress tensor to be skew symmetric. Furthermore, the mean curvature tensor, which is the skew symmetric part of the gradient of macrorotations, is shown to be the correct energy conjugate of couple stress (Hadjesfandiari and Dargush (2011)).

In the past few years, there has been an increasing use of macro-rotation-based couple stress theories. Most of the applications are based on Yang et al. (2002), which also is known as modified couple stress theory. Romanoff and Reddy (2014) did an experimental study of web core sandwich panels and compared results with modified couple stress theory for macro-scale Timoshenko beams. Mohammadi et al. (2017) studied the effect of modified couple stress theory on conical nanotubes and compared results with molecular dynamics simulations. Tan and Chen (2019) carried out size-dependent electro-thermo-mechanical analysis of multilayer cantilever microactuators by Joule heating using modified couple stress theory. Lata and Kaur (2019a, b) studied deformation in a transversely isotropic thermoelastic medium using new modified couple stress theory, more closely related to the skew-symmetric couple stress theory. In addition, there has been increasing direct use of consistent couple stress theory by Hadjesfandiari and Dargush (2011). For example, Li et al. (2014) carried out analysis on three-layer microbeams, including electromechanical coupling using consistent couple stress theory, while Dehkordi and Beni (2017) studied electromechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory and compared it with molecular dynamics simulations. Patel et al. (2017) presented simple moment-curvature approach for large deflection analysis of microbeams using consistent couple stress theory. Subramaniam and Mondal (2020) studied the effect of couple stresses on the rheology and dynamics of linear Maxwell viscoelastic fluids. It is important to note that modifed couple stress theory and consistent couple stress theory become equivalent for beam and in-plane deflection problems. According to the best of our knowledge, there is no proper experimental validation of any particular theory. In any case, the debate on correctness of theories is beyond the scope of the present paper, as the current work deals with the development of an effective computational method to study the effect of skew-symmetric couple stress in anisotropic materials.

In this present work, a finite element method based on the skew symmetric couple stress theory (Hadjesfandiari and Dargush (2011)) is developed. This couple stress theory is a fourth order theory. Upon creating the variational formulation, we are left with second order derivatives of displacements. This means a fully displacement based finite element method (FEM) for couple stress theory would require $C^1$ continuity. To reduce the challenge in maintaining $C^1$ continuity, three methods have been developed in the past that require at most $C^0$ continuity. Darrall et al. (2014) defined displacements and rotations as independent variables and then used Lagrange multipliers to constrain these rotations to one half the curl of displacements. Chakravarty et al. (2017) also defined displacements and rotations as independent variables and used penalty parameters to constrain rotations to the displacements. On the other hand, Deng and Dargush (2017) developed a mixed variational formulation with displacements, stresses and couple stresses as independent variables for couple stress elastodynamics. In recent years many developments have been happening in the field of mixed variational methods, which differ from traditional displacement based formulations by including independent variables, such as stresses, strains and surface tractions. The first development can be traced back to the famous Reissner (1950) and Hu (1955); Washizu (1975) principles. Recently, a number of researchers (Sivaselvan and Reinhorn (2006); Sivaselvan et al. (2009); Lavan et al. (2009); Apostolakis and Dargush (2011); Lavan (2010); Apostolakis and Dargush (2013)) have used mixed variational formulations to solve some interesting and challenging problems in engineering. Also, some analyses of other size dependent theories, namely, micropolar (Sachio et al. (1984); Ghosh and Liu (1995); Huang et al. (2000); Providas and Kattis (2002); Li and Xie (2004); Sharbati and Naghdabadi (2006); Riahi and Curran (2009)), strain gradient (Chen and Wang (2002); Wei (2006)) and couple stress (Wood (1988); Ma et al. (2008); Reddy (2011)) have been done using mixed variational methods.

The method presented in the current work is a mixed formulation based on Deng and Dargush (2017) with a slightly different representation. Here, the novelty involves the use of only two polar (true) vectors, displacement and couple stress, as primary variables. We apply the resulting stationary principle and finite element method to solve consistent couple stress problems in linear anisotropic elasticity for the first time. For isotropic materials, we have two independent parameters in the constitutive relations between force stresses and strains and one additional parameter in the couple stress and curvature relations (Hadjesfandiari and Dargush (2011)). For anisotropic materials, these parameters will be more numerous and interestingly there might be coupling present between force stress - curvatures and similarly in couple stresses - strains. These coupling constitutive relations involve a third order tensor. However, as shown in Nye (1985), a third order tensor has non-zero entries only for non-centrosymmetric materials. Therefore, it is very important to classify materials into centrosymmetric and non-centrosymmetric categories for anisotropic couple stress elasticity. The focus of the current work is restricted to the centrosymmetric category.

The organization of this paper is as follows. An overview of the governing equations, which are required for FEM is presented in Section 2. This basically involves important parts of kinematics, kinetics, boundary conditions and constitutive relations taken from Hadjesfandiari and Dargush (2011). Section 3 concentrates on centrosymmetric materials and incorporates the development of variational formulations. Section 4 then presents the corresponding finite element method. Computational examples are presented in Section 5 to show the effects of this couple stress theory. Finally, in Section 6, conclusions are presented, followed by future work.